In [1]}, we called a binary relational structure \(R\)  minimal prime if \(R\)  is prime and every prime induced structure with the same cardinality as \(R\)  embeds a copy of \(R\) . We described the countable minimal prime graphs with no infinite clique (Theorem 2, p.358). From our description follows that a countable graph \(G\)  which is the incomparability graph of a poset coverable by two chains is minimal prime if \(G\)  is either isomorphic to the one way infinite path \(\mathrm {P}_{\infty }\)  or to the half-complete bipartite graph \(H\) . Thus, \(Age(G)\)  is minimal prime. The question to know wether or not the age of a relational structure is minimal prime whenever \(R\)  is minimal prime was considered by Oudrar ([2]} Problème 1, page 99). Her and our results suggest that the answer is positive. But, a description of minimal prime bichains or posets is not known; the same question for minimal prime ages is unsolved too.
